Question

A man wants to reach point $B$ on the opposite bank of a river flowing at a speed $u$ as shown in Fig. What minimum speed relative to water should the man have so that he can reach directly to point $B$? In which direction should be swim?

Answer 1

${\overline{v}}_m={\overline{v}}_{m/w}+{\overline{v}}_w$

$=v\cos (\theta +{45}^o)\hat{i}+v\sin (\theta +{45}^o)\hat{j}+u\cos {45}^o\hat{i}-usin{45}^o\hat{j}$

$=\left[v\cos \right(\theta +{45}^o)+u\cos {45}^o]\hat{i}+\left[v\sin \right(\theta +{45}^o-u\sin {45}^o]\hat{j}$

to reach the point $B$ $\hat{j}$ component should be zero

$v\sin (\theta +{45}^o)=u\sin {45}^o$

$\Rightarrow v=\frac{u\sin {45}^o}{\sin (\theta +{45}^o)}$

For $v_{min},\theta +{45}^o={90}^o\Rightarrow \theta ={45}^o$

$\to v_{min}=\frac{u}{\sqrt{2}},{45}^o$ with upstream